I proved that $latex \lvert \mathbb{R} \rvert_{\geq} \neq \lvert \mathbb{R} \rvert \sqcap \geq$ and so disproved one of my conjectures.

The proof is currently available in the section “Some inequalities” of this PDF file.

The proof isn’t yet thoroughly checked for errors.

Note that I have not yet proved $latex \lvert \mathbb{R} \rvert_{>} \neq \lvert \mathbb{R} \rvert \sqcap >$, but the proof is expected to be similar to the above.

“The proof isn’t yet thoroughly checked for errors.” Yes, and I found an error in the proof. I am now working on correcting this error.